This article is from the rec.audio.* FAQ, by with numerous contributions by Bob Neidorff others.
The speeds were chosen because that is the speed that resulted
when you used standard parts. Electric motors rotate at 1800
rpm, most shafts are 1/4". Those combinations with the proper
gears and idlers came out to 78 rpm. In reality it's 78.26
rpm. Tape recorder speeds evolved the same way.
The 78.26 was standardized after electric recording/playback
occured. Prior to that, speeds were "in the neighborhood of"
78 rpm. Some lower and some higher. 80 rpm was used in many
recordings. (Courtesy of Bill Vermillion)